Question

The sum of four numbers A, B, C, and D is 10,600. Without A, the average of B, C, and D is 1,000. Without B, the average of A, C, and D is 3,220. Without C, the average of A, B, and D is 3,180. Find the value of D.

• A. 880
• B. 7600
• C. 1000
• D. 1120

Hint:

Questions involving averages of subsets are easiest to solve by converting every average into a sum.

Answer 1

The Correct Option is C

Step 1: Use the given averages. Without A, \[ \frac{B+C+D}{3}=1000 \] \[ B+C+D=3000 \] Hence, \[ A=10600-3000=7600 \] Without B, \[ \frac{A+C+D}{3}=3220 \] \[ A+C+D=9660 \] Substituting \(A=7600\), \[ C+D=2060 \] Without C, \[ \frac{A+B+D}{3}=3180 \] \[ A+B+D=9540 \] Substituting \(A=7600\), \[ B+D=1940 \]

Step 2: Find D. From \[ B+C+D=3000 \] Using \[ C+D=2060 \] we obtain \[ B=940 \] Now, \[ B+D=1940 \] \[ 940+D=1940 \] \[ D=1000 \] Hence, \[ \boxed{D=1000} \] Therefore, \[ \boxed{\text{Answer = (C)}} \]